A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.
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37 views
Stochastic process, Gaussian, with zero mean is a Wiener process
Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
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25 views
Stopping times and $\sigma$-algebras
We have the usual $(\Omega, \mathcal{F}, P)$ stochastic basis. Let $\rho, \tau: \Omega \to T \cup \{+\infty\}$ be stopping times and $\mathcal{F}_{\rho}, \mathcal{F}_{\tau}$ their respective ...
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2answers
40 views
Chaos in finite field
Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map
$x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $
where $\mathcal{P}$ - ...
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0answers
25 views
Prove $\mathbb{E}[X_t | \mathcal{F}_s] = \mathbb{E}[X_t | \sigma(\mathcal{F}_s \cup \mathcal{G}_s)] $
We want to prove that if $X_t$ is an $\mathcal{F}_t$ - martingale: $\mathbb{E}[X_t | \mathcal{F}_s] = X_s$ for $s<t$, then it's also a $\sigma(\mathcal{F}_s \cup \mathcal{G}_s)$- martingale. ...
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0answers
12 views
Levy process absolute moment
For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
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0answers
30 views
Construction binary tree
First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$.
We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$
...
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1answer
26 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
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10 views
Analytic tools in the theory of Galton-Watson processes
The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
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1answer
36 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
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2answers
27 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
3
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0answers
28 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
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17 views
Show $B_{t}^{2}$ is a weak solution of a stochastic differential equation. [closed]
Let $B_{T}$ be a Brownian motion in $\mathbb{R}$.
Show that $X_{t} = B_{t}^{2}$ is a weak solution of the stochastic differential equation
$dX_{t} = dt + 2\sqrt{|X_{t}|}d\tilde{B_{t}}$
where ...
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0answers
39 views
Exponential Levy process
We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process
...
3
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1answer
63 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
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1answer
39 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
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2answers
51 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
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10 views
Distribution of partial sums of a $L^2$-transformed Gaussian Process
Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
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1answer
21 views
Proving weak existence of CIR process
Consider the following SDE
$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$
where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
1
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1answer
45 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
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1answer
20 views
Can an absorbing CTMC be reversible?
Can a CTMC with an absorbing state be reversible? I guess not, as the product of rates through any loop cannot be equal when the loop involves the absorbing state (Kolmogorov criterion). Is my ...
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1answer
34 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
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1answer
35 views
Finding stationary Distribution
I need to know how to find the stationary distribution for this matrix:
$$
Q=
\begin{bmatrix}
-2 & 2 & 0 & 0 \\
1 & -2 & 1 & 0 \\
0 & 1 & -2 &1\\
0 & 0 ...
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1answer
66 views
Canonical Markov Process
Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel-$\sigma$-algebra $\mathcal{E}$ and we assume that $t\rightarrow E_{X_{t}}f(X_{s})$ ...
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22 views
Girsanov kernel moments
Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
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0answers
20 views
Linear birth death process extinction probabilities
Given a birth and death process $X$ with $\lambda_n=n\lambda$ and $\mu_n=n\mu$ for $n\ge0$, and letting $P_n(t)=\Pr\{X(t)=n\}$, I need to prove that $P_0'(t)=\lambda P_0(t)^2-(\lambda+\mu)P_0(t)+\mu$.
...
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1answer
36 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
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0answers
13 views
Solving the following SDE: dS=S(μdt+σe^(-t)dZ) from the BS-Model
I am trying to do an exercise where I have to solve the following stochastic Differential Equation, which is described by a modification of the Black-Scholes Model. It looks like the folllowing:
...
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1answer
26 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
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13 views
fGn asymptotic claim correlation
Let $(X_{i})$ be the fractional Gaussian noise for $H\in(0,1)$.
Since it is stationary $\mathbb{E}(X_{i}X_{j})$ only depends on $|j-i|$.
How can I prove for $\rho(|j-i|)=\mathbb{E}(X_{i}X_{j})$ that ...
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1answer
39 views
Moment generating function of two non-independent Brownian increments
I am writing to ask if it is possible to get closed-form solution to the expression to the following expression:
$\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
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0answers
21 views
Random process x(t) =C and C is uniform over [-2,3]
I need reassurance that if I do a a few sample realizations of this random process they are all going to look the same. They are going to be an horizontal line with x(t) constant equal to 1/5.
I see ...
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0answers
55 views
Graduate research project in stochastic programming . [closed]
I don't know is this a good question or is this place is right to post this like question or not , but I need keen help, so I'm posting it.
I'm a graduate student & in this semester I've ...
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1answer
22 views
Integrating a Poisson Process with respect to itself
I am just learning about Poisson Processes and I feel somewhat comfortable with the basic concepts, but I am a little stuck with the following problem:
Let $N(t)$ be a Poisson process with intensity ...
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57 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
3
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0answers
26 views
Orthogonal projections for minimization problem
I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
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0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
2
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0answers
40 views
Pure Birth Process
I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on ...
2
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0answers
51 views
Progressive measurability of stopped process
Let $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$ be a filtration and let $X$ be a stochastic process progressively measurable with respect to $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$. Let $T$ be a stopping time ...
2
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0answers
35 views
Product of predictable process and a characteristic function is integrable
Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that
$$\int_0^T\theta_u dS_u\ge -a$$
for a $a>0$. Furthermore ...
1
vote
2answers
68 views
Standard Brownian Motion
Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$.
Can anyone give me some hint to start the ...
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0answers
14 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
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1answer
51 views
Rewriting Markov process
Let $X$ be a Markov proces with state space $(E,\mathcal{E})$with initial distribution $\nu$ and transition function $P_{t}$, so $$E_{\nu}(f(X_{t+s})\mid\mathcal{F}_{s})=P_{t}f(X_{s})$$
Suppose that ...
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1answer
101 views
Show that $M$ is a martingale
Let $B$ be typical Brownian motion with $\mu >0$ and $x \in \mathbb{R}$. $X(t):=x+B(t)+\mu t$, for each $t\geqslant 0$, Brownian motion with velocity $\mu$ that starts at $x$. For $r \in ...
1
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1answer
80 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
3
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0answers
58 views
Inadmissibility of Simpson's rule
Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation
$$
\mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, ...
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23 views
White Noise Process
Suppose $w_{t}$ is a normal white noise process. Is $z_{t} = w_{t}*w_{t-1}$ stationary?
Is my reasoning correct?
$Ew_{t}w_{t-1}w_{t+h}w_{t+h-1} = 0 $
for all $h$ implying that the series is ...
1
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0answers
21 views
Stochastic Processes, Doob's Inequality
please confirm if the following is correct.
Let $A = \{V_\tau > \epsilon\}$ and $\alpha = \min(\alpha_1,\tau)$, where $\alpha_1 = \min\{t\geq 0: X_t\geq \epsilon\}$ (So intuitively, $\alpha$ ...
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1answer
32 views
Determining Stationarity of Sine Process
Suppose $X_{t} = B\sin(\omega t) + w_{t}$ and $\omega$ is between $(0,\pi)$, $B$ has mean 0 and variance 1 $w_{t}$ is $N(0,1)$ and $w_{t}$ is independent of $B$. Show that $X_{t}$ is weakly ...
2
votes
1answer
65 views
Sum of stationary process
Suppose you have two stationary process $A_{t}$ and $B_{t}$. Suppose $Z_{t} = A_{t} + B_{t}$. Show that $Z_{t}$ is stationary. I am unsure how to solve this without knowing if the processes are ...
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1answer
38 views
Product of stationary stochastic process
Suppose $z_{t} = x_{t}y_{t}$ where $x_{t}$ and $y_{t}$ are 0 mean, independent stationary stochastic process. What is the autocovariance function of $z_{t}$? Show that the spectral density can be ...
